Cluster analysis in R determine the optimal number of clusters

Question

Being a newbie in R  I m not very sure how to choose the best number of clusters to do a k-means analysis  After plotting a subset of below data  how many clusters will be appropriate  How can I perform cluster dendro analysis   n   1000 kk   10     x1   runif kk  y1   runif kk  z1   runif kk      x4   sample x1 length x1   y4   sample y1 length y1    randObs  lt - function       ix   sample  1 length x4   1     iy   sample  1 length y4   1     rx   rnorm  1  x4 ix   runif 1  8     ry   rnorm  1  y4 ix   runif 1  8     return  c rx ry        x   c   y   c   for   k in 1 n       rPair     randObs     x     c  x  rPair 1      y     c  y  rPair 2      z  lt - rnorm n  d  lt - data frame  x  y  z

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If your question is how can I determine how many clusters are appropriate for a kmeans analysis of my data   then here are some options  The wikipedia article on determining numbers of clusters has a good review of some of these methods   First  some reproducible data  the data in the Q are    unclear to me    n   100 g   6  set seed g  d  lt - data frame x   unlist lapply 1 g  function i  rnorm n g  runif 1  i 2                      y   unlist lapply 1 g  function i  rnorm n g  runif 1  i 2     plot d      One  Look for a bend or elbow in the sum of squared error  SSE  scree plot  See http   www statmethods net advstats cluster html  amp  http   www mattpeeples net kmeans html for more  The location of the elbow in the resulting plot suggests a suitable number of clusters for the kmeans   mydata  lt - d wss  lt -  nrow mydata -1  sum apply mydata 2 var     for  i in 2 15  wss i   lt - sum kmeans mydata                                         centers i  withinss  plot 1 15  wss  type  b   xlab  Number of Clusters        ylab  Within groups sum of squares     We might conclude that 4 clusters would be indicated by this method    Two  You can do partitioning around medoids to estimate the number of clusters using the pamk function in the fpc package    library fpc  pamk best  lt - pamk d  cat  number of clusters estimated by optimum average silhouette width    pamk best nc    n   plot pam d  pamk best nc          we could also do  library fpc  asw  lt - numeric 20  for  k in 2 20    asw  k    lt - pam d  k    silinfo   avg width k best  lt - which max asw  cat  silhouette-optimal number of clusters    k best    n     still 4   Three  Calinsky criterion  Another approach to diagnosing how many clusters suit the data  In this case we try 1 to 10 groups   require vegan  fit  lt - cascadeKM scale d  center   TRUE   scale   TRUE   1  10  iter   1000  plot fit  sortg   TRUE  grpmts plot   TRUE  calinski best  lt - as numeric which max fit results 2     cat  Calinski criterion optimal number of clusters    calinski best    n     5 clusters      Four  Determine the optimal model and number of clusters according to the Bayesian Information Criterion for expectation-maximization  initialized by hierarchical clustering for parameterized Gaussian mixture models    See http   www jstatsoft org v18 i06 paper   http   www stat washington edu research reports 2006 tr504 pdf   library mclust    Run the function to see how many clusters   it finds to be optimal  set it to search for   at least 1 model and up 20  d clust  lt - Mclust as matrix d   G 1 20  m best  lt - dim d clust z  2  cat  model-based optimal number of clusters    m best    n     4 clusters plot d clust        Five  Affinity propagation  AP  clustering  see http   dx doi org 10 1126 science 1136800  library apcluster  d apclus  lt - apcluster negDistMat r 2   d  cat  affinity propogation optimal number of clusters    length d apclus clusters     n     4 heatmap d apclus  plot d apclus  d       Six  Gap Statistic for Estimating the Number of Clusters  See also some code for a nice graphical output  Trying 2-10 clusters here   library cluster  clusGap d  kmeans  10  B   100  verbose   interactive     Clustering k   1 2      K max    10      done Bootstrapping  b   1 2      B    100    one     per sample                                                      50                                                     100  Clustering Gap statistic   clusGap    B 100 simulated reference sets  k   1  10  -- gt  Number of clusters  method  firstSEmax   SE factor 1   4           logW   E logW        gap     SE sim   1   5 991701 5 970454 -0 0212471 0 04388506   2   5 152666 5 367256  0 2145907 0 04057451   3   4 557779 5 069601  0 5118225 0 03215540   4   3 928959 4 880453  0 9514943 0 04630399   5   3 789319 4 766903  0 9775842 0 04826191   6   3 747539 4 670100  0 9225607 0 03898850   7   3 582373 4 590136  1 0077628 0 04892236   8   3 528791 4 509247  0 9804556 0 04701930   9   3 442481 4 433200  0 9907197 0 04935647  10   3 445291 4 369232  0 9239414 0 05055486   Here s the output from Edwin Chen s implementation of the gap statistic    Seven  You may also find it useful to explore your data with clustergrams to visualize cluster assignment  see http   www r-statistics com 2010 06 clustergram-visualization-and-diagnostics-for-cluster-analysis-r-code  for more details    Eight  The NbClust package provides 30 indices to determine the number of clusters in a dataset   library NbClust  nb  lt - NbClust d  diss NULL  distance    euclidean           method    kmeans   min nc 2  max nc 15           index    alllong   alphaBeale   0 1  hist nb Best nc 1    breaks   max na omit nb Best nc 1        Looks like 3 is the most frequently determined number of clusters   and curiously  four clusters is not in the output at all      If your question is how can I produce a dendrogram to visualize the results of my cluster analysis  then you should start with these  http   www statmethods net advstats cluster html http   www r-tutor com gpu-computing clustering hierarchical-cluster-analysis http   gastonsanchez wordpress com 2012 10 03 7-ways-to-plot-dendrograms-in-r  And see here for more exotic methods  http   cran r-project org web views Cluster html  Here are a few examples   d dist  lt - dist as matrix d       find distance matrix  plot hclust d dist               apply hirarchical clustering and plot       a Bayesian clustering method  good for high-dimension data  more details    http   vahid probstat ca paper 2012-bclust pdf install packages  bclust   library bclust  x  lt - as matrix d  d bclus  lt - bclust x  transformed par   c 0  -50  log 16   0  0  0   viplot imp d bclus  var   plot d bclus   ditplot d bclus  dptplot d bclus  scale   20  horizbar plot   TRUE varimp   imp d bclus  var  horizbar distance   0  dendrogram lwd   2    I just include the dendrogram here     Also for high-dimension data is the pvclust library which calculates p-values for hierarchical clustering via multiscale bootstrap resampling  Here s the example from the documentation  wont work on such low dimensional data as in my example     library pvclust  library MASS  data Boston  boston pv  lt - pvclust Boston  plot boston pv      Does any of that help

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The answers are great  If you want to give a chance to another clustering method you can use hierarchical clustering and see how data is splitting    gt  set seed 2   gt  x matrix rnorm 50 2   ncol 2   gt  hc complete   hclust dist x   method  complete    gt  plot hc complete      Depending on how many classes you need you can cut your dendrogram as    gt  cutree hc complete k   2    1  1 1 1 2 1 1 1 1 1 1 1 1 1 2 1 2 1 1 1 1 1 2 1 1 1  26  2 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 2 1 1 1 1 1 1 1 2   If you type  cutree you will see the definitions  If your data set has three classes it will be simply cutree hc complete  k   3   The equivalent for cutree hc complete k   2  is cutree hc complete h   4 9

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These methods are great but when trying to find k for much larger data sets  these can be crazy slow in R   A good solution I have found is the  RWeka  package  which has an efficient implementation of the X-Means algorithm - an extended version of K-Means that scales better and will determine the optimum number of clusters for you   First you ll want to make sure that Weka is installed on your system and have XMeans installed through Weka s package manager tool   library RWeka     Print a list of available options for the X-Means algorithm WOW  XMeans      Create a Weka control object which will specify our parameters weka ctrl  lt - Weka control      I   1000                             max no  of overall iterations     M   1000                             max no  of iterations in the kMeans loop     L   20                               min no  of clusters     H   150                              max no  of clusters     D    weka core EuclideanDistance     distance metric Euclidean     C   0 4                              cutoff factor         S   12                               random number seed  for reproducibility       Run the algorithm on your data  d x means  lt - XMeans d  control   weka ctrl     Assign cluster IDs to original data set d xmeans cluster  lt - x means class ids

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A simple solution is the library factoextra  You can change the clustering method and the method for calculate the best number of groups  For example if you want to know the best number of clusters for a k- means   Data  mtcars  library factoextra     fviz nbclust mtcars  kmeans  method    wss           geom vline xintercept   3  linetype   2         labs subtitle    Elbow method     Finally  we get a graph like

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In order to determine optimal k-cluster in clustering methods  I usually using Elbow method accompany by Parallel processing to avoid time-comsuming  This code can sample like this   Elbow method  elbow k  lt - function mydata   dist obj  lt - dist mydata  hclust obj  lt - hclust dist obj  css obj  lt - css hclust dist obj hclust obj  elbow obj  lt - elbow batch css obj  k  lt - elbow obj k return k      Running Elbow parallel  no cores  lt - detectCores       cl lt -makeCluster no cores      clusterEvalQ cl  library GMD       clusterExport cl  list  data clustering    data convert    elbow k    clustering kmeans     start time  lt - Sys time    elbow k handle data clustering    k clusters  lt - parSapply cl  1  function x  elbow k data clustering       end time  lt - Sys time       cat  Time to find k using Elbow method is   end time - start time   seconds with k value    k clusters    It works well

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It s hard to add something too such an elaborate answer  Though I feel we should mention identify here  particularly because  Ben shows a lot of dendrogram examples    d dist  lt - dist as matrix d       find distance matrix  plot hclust d dist    clusters  lt - identify hclust d dist     identify lets you interactively choose clusters from an dendrogram and stores your choices to a list  Hit Esc to leave interactive mode and return to R console  Note  that the list contains the indices  not the rownames  as opposed to cutree

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Splendid answer from Ben  However I m surprised that the Affinity Propagation  AP  method has been here suggested just to find the number of cluster for the k-means method  where in general AP do a better job clustering the data  Please see the scientific paper supporting this method in Science here   Frey  Brendan J   and Delbert Dueck   Clustering by passing messages between data points   science 315 5814  2007   972-976   So if you are not biased toward k-means I suggest to use AP directly  which will cluster the data without requiring knowing the number of clusters   library apcluster  apclus   apcluster negDistMat r 2   data  show apclus    If negative euclidean distances are not appropriate  then you can use another similarity measures provided in the same package  For example  for similarities based on Spearman correlations  this is what you need   sim   corSimMat data  method  spearman   apclus   apcluster s sim    Please note that those functions for similarities in the AP package are just provided for simplicity  In fact  apcluster   function in R will accept any matrix of correlations  The same before with corSimMat   can be done with this   sim   cor data  method  spearman     or   sim   cor t data   method  spearman     depending on what you want to cluster on your matrix  rows or cols

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